# Hypothesis Testing

An obvious question is how to carry out various diagnostic tests done in the parametric econometrics within the nonparametric and semiparametric models. Several papers have appeared in the recent literature which deal with this issue. We present them here and show their links.

First consider the problem of testing a specified parametric model against a nonparametric alternative, H0 : /(p, x) = E(yi | xi) against H1 : m(x) = E(yi | xi). The idea behind the Ullah (1985) test statistic is to compare the parametric RSS (PRSS) XЙ2, x = Vi – /(S, xi) with the nonparametric RSS (NPRSS), XB2, where щ = – m(xi). His test statistic is

or simply T * = (PRSS – NPRSS), and reject the null hypothesis when T1 is large. л/П T1 has a degenerage distribution under H0. Lee (1994) uses density weighted

residuals and compares Xwfi} with X A2 to avoid degeneracy, for other procedures see Pagan and Ullah (1999). Pagan and Ullah (1999) also indicate the normalizing factor needed for the asymptotic normality of T1. An alternative suggested here is the following nonparametric bootstrap method:

1. Generate the bootstrap residuals u* from the centered residuals (fl; – u) where й is the average of й.

2. Generate y* = f (S, x,) + u* from the null model.

3. Using the bootstrap sample x;, y*, i = 1,…, n, estimate m(x) nonparametrically, say and m*(x,), and obtain the bootstrap residual A* = y* – m*(x,).

4. Calculate the bootstrap test statistic T* = (Xй2 – Xй*)2/Хй*2.

5. Repeat steps (1) to (4) B times and use the empirical distribution of T* as the null distribution of T*.

An alternative is to use wild bootstrap method or pivotal bootstrap which will preserve the conditional heteroskedasticity in the original residuals. Another alternative is to use the block bootstrap method (Buhlman and Kunsch, 1995).

An alternative test statistic is based on comparing the parametric fit with the nonparametric fit. Defining a(x) as a smooth weight function, this test statistic is

(f (S, x,) – m(x,))2a(x,)dF(x),

where F is the empirical distribution function, see Ullah (1985) and Gozalo

(1995) , and Ai’t-Sahalia et al. (1998) who also indicate that f(S, x,) and m(x,) can also be replaced by f(S, x,) – ff (S, x,)f(x,)dxi and m(xi) by m(xi) – fm(x,)f(x)dxi without affecting the results in practice and provide the asymptotic normality of

nhq/2T2.

Hardle and Mammen (1993) suggest a weighted integrated square difference between the nonparametric estimator and the kernel smoothed parametric estimator f ф, x,) = E(f(S, xf)| x,-) which can be calculated by the LLS procedures with y, replaced by f (S, x,). This is

where E(m|x) = E( y|x) – E(f(S, x)|x) = m(x) – f(S, x). It has been shown in Rahman and Ullah (1999) that the use of the kernel smoothed estimator f(S, x) gives better size and power performances of the tests compared to the case of using f(S, x). T3 is similar to T2 if we write a(x) = a(x)f-1(x)dF(x) and use the empirical distribution. This test statistic is computationally involved. In view of this, Li and Wang (1998) and Zheng (1996) proposed a conditional moment test (CMT) which is easy to calculate, and has a better power performance. Its form is

1 n

X Ui(m(x,) – x!))f(x!).

n i

This statistic is based on the idea that under the null E(uixl) = E[uiE(uix)] = E[(Em;|x;)2] = E[uiE(uix)a(x)] = 0 for any positive a(x). T4 is an estimate of E[u;E(u; x;)a(x;)] for a(x,) = f(x). Eubank and Spiegelman (1990), however, tests for E[(Eu; x;)2] = 0 using a series type estimator of E(u; x;). The test statistic T4 has nhq/2 rate of convergence to normality.

All the above nonparametric tests are generally calculated with the leave-one – out estimators of m(xi) = m-i(x) and the weight a(x.) = f(xi) = f-i(xI). Theoretically, the use of leave-one-out estimators helps to get asymptotic normality centered at zero. The tests are consistent model specification tests in the sense that their power goes to one as n ^ ™ against all the alternatives. The usual parametric specification tests are however consistent against a specified alternative. An approach to developing consistent model specification test, without using any nonparametric estimator of m(x), is the CMT due to Bierens-Newey-Tauchen, see Pagan and Ullah (1999) for details. An important difference between Bierens – type tests and the tests T1 to T6 is the treatment of h. While T1 to T6 tests consider h ^ 0 as n ^ ж, Bierens (1982) type tests treat h to be fixed which makes the asymptotic distribution of their tests to be nonnormal but can detect the Pitman’s

local alternative that approach the null at the rate O(n~1/2) compared to the slower rate of O((nhq/2)-1/2) of T1 to T6. However, Fan and Li (1996) indicate that under high frequency type local alternatives the tests with vanishing h may be more powerful than tests based on fixed h. For asymptotic normality of the tests T1 to T6 for the dependent observations, see Li (1997).

The test statistics T1 to T6 can also be used for the problem of variable selections. For example, testing H0 : m(x,) = m(xi1, xi2) = m(xi1) against H1 : m(x) Ф m(xa) can be carried out by calculating the difference between the NPRSS due to m(xi1, x12) and the NPRSS due to m(xi1), or using T2 = n-1’L(m(xi1, xi2) – m(xi1))2a(x,) test, see Ai’t-Sahalia et al. (1998) for asymptotic normality. An alternative is suggested in Racine (1997). We can also do the diagnostics for variable selection by using the goodness of fit measures described in Section 2.1; in addition see Vien (1994) where the cross-validation method has been used.

The tests T1 to T6 can also be extended to do nonnested testing (Delgado and Mora, 1998), testing for parametric and semiparametric models yi = f ф, x;) + Xm(xJ) + ui, (Li, 1997; Fan and Li, 1996) and single index models, Ai’t-Sahalia et al., 1998). Finally, for testing the restrictions on the parameters, testing hetero – skedasticity, and testing serial correlation in the parametric model yi = f(P, x;) + ui with the unknown form of density, see Gonzalez-Rivera and Ullah (1999) where they develop the semiparametric Rao-score test (Lagrange multiplier) with the unknown density replaced by its kernel estimator. Also see Li and Hsiao (1998) for a semiparametric test of serial correlation.

Note

* The author is thankful to two referees for their constructive comments and suggestions. The research support from the Academic Senate, UCR, is gratefully acknowledged.

References

Ai, C. (1997). A semiparametric maximum likelihood estimator. Econometrica 65, 933-63.

A’lt-Sahalia, Y., P. J. Bickel, and T. M. Stoker et al. (1998). Goodness-of-fit regression using kernel methods. Manuscript, University of Chicago.

Anglin, P., and R. Gencay (1996). Semiparametric estimation of a hedonic price function. Journal of Applied Econometrics 11, 633-48.

Bierens, H. J. (1982). Consistent model specification tests. Journal of Econometrics 20, 105-34.

Breiman, L., and J. Friedman (1985). Estimating optimal transformations for multiple regression and correlation. Journal of the American Statistical Association 80, 580-619.

Buhlman, P., and H. R. Kunsch (1995). The blockwise bootstrap for general parameters of a stationary time series. Scandinavian Journal of Statistics 22, 35-54.

Chen, R., W. Hardle, O. B. Linton, and E. Sevarance-Lossin (1996). Nonparametric estimation of additive separable regression model. Statistical Theory and Computational Aspect of Smoothing; Physica-Verlag 247-65.

Cai, Z., J. Fan, and Q. Yao (1998). Functional-coefficient regression models for non-linear time series. Manuscript, University of North Carolina.

Delgado, M. A., and J. Mora (1998). Testing non-nested semiparametric models: An application to Engel curve specification. Journal of Applied Econometrics 13, 145-62.

Engel, R. F., C. W.J. Granger, J. Rice, and A. Weiss (1986). Semiparametric estimates of the relation between weather and electricity sales. Journal of the American Statistical Association 81, 310-20.

Eubank, R. L., and C. H. Spiegelman (1990). Testing the goodness-of-fit of the linear models via nonparametric regression techniques. Journal of the American Statistical Association 85, 387-92.

Fan, J., and I. Gijbels (1996). Local Polynomial Modelling and Its Applications. London: Chapman and Hall.

Fan, Y., and Q. Li (1996). Consistent model specification tests: Omitted variables and semiparametric functional forms. Econometrica 64, 865-90.

Fan, Y., and Q. Li (1997). On estimating additive partially linear models. Manuscript, University of Windsor.

Fan, Y., and A. Ullah (1998). Asymptotic normality of a combined regression estimator. Journal of Multivariate Analysis 71, 191-240.

Fan, Y., and A. Ullah (1999). On goodness-of-fit tests for weakly dependent processes using kernel method. Journal of Nonparametric Statistics 11, 337-60.

Faraway, J. (1990). Bootstrap selection for bandwidth and confidence bands for nonparametric regression. Journal of Statistics Computational Simulations 37, 37-44.

Friedman J. H., and J. W. Tukey (1974). A projection pursuit algorithm for exploratory data analysis. IEEE. Transactions on Computers, C-23, 881-90.

Glad, I. K. (1998). Parametrically guided nonparametric regression. Scandinavian Journal of Statistics 25, 649-68.

Gozalo, P. L. (1995). Nonparametric specification testing with – Jn-local power and bootstrap critical values. Working Paper no. 95-21R; Brown University.

Gonzalez-Rivera, G., and A. Ullah (1999). Rao’s score test with nonparametric density estimators. Journal of Statistical Planning and Inference.

Hardle, W. (1990). Applied Nonparametric Regression. New York: Cambridge University Press.

Hardle, W., and A. Bowman (1988). Bootstrapping in nonparametric regression: Local adaptive smoothing and confidence bounds. Journal of the American Statistical Association 83, 102-10.

Hardle, W., P. Hall, and J. S. Marron (1992). Regression smoothing parameters that are not far from their optimum. Journal of the American Statistical Association 87, 277-33.

Hardle, W., and E. Mammen (1993). Comparing nonparametric versus parametric regression fits. Annals of Statistics 21, 1926-47.

Hastie, T., and R. Tibshirani (1990). General Additive Models. New York: Chapman and Hall.

Herrman, E., T. Gasser, and A. Kneip (1992). Choice of bandwidth for kernel regression when residuals are correlated. Biometrika 79, 783-95.

Kuan, C. M., and H. White (1994). Artificial neural networks: An econometric perspective. Econometric Reviews 13, 1-91.

Lee, B. J. (1994). Asymptotic distribution of the Ullah-type against the nonparametric alternative. Journal of Quantitative Economics 10, 73-92.

Lewbel, A., and O. Linton (1998). Nonparametric censored regression. Manuscript, no. 1186, Yale University.

Li, Q. (1997). Consistent model specification tests for time series models. Manuscript, University of Guelph.

Li, Q., and C. Hsiao (1998). Testing serial correlation in semiparametric panel data models. Journal of Econometrics 87, 207-37.

Li, Q., and T. Stengos (1996). Semiparametric estimation of partially linear panel data models. Journal of Econometrics 71, 389-97.

Li, Q., and A. Ullah (1998). Estimating partially linear panel data models with one way error components. Econometric Reviews 17, 145-66.

Li, Q., and S. Wang (1998). A simple consistent bootstrap test for a parametric regression function. Journal of Econometrics 87, 145-65.

Li, Q., C. Huang, and T. T. Fu (1998). Semiparametric smooth coefficient stochastic frontier models. Manuscript, Institute of Economics, Taiwan.

Linton, O., and D. Nielson (1995). Estimating structured nonparametric regression by the kernel method. Biometrika 82, 93-101.

Linton, O., and P. L. Gozalo (1996). Testing additivity in generalized nonparametric regression models. Working Paper, Yale University and Brown University.

Nadaraya, E. A. (1964). On estimating regression. Theory of Probability and its Applications 9, 141-2.

Newey, W. K. (1985). Maximum likelihood specification testing and conditional moment tests. Econometrica 53, 1047-70.

Newey, W. K. (1997). Convergence rates and asymptotic normality of series estimators. Journal of Econometrics 29, 147-68.

Pagan, A. R., and A. Ullah (1999). Nonparametric Econometrics. Cambridge: Cambridge University Press.

Phillips, P. C.B., and J. Y. Park (1998). Nonstationary density estimation and kernel autoregression. Manuscript, no. 1181, Yale University.

Powell, J. L., H. Stock, and T. M. Stoker (1989). Semiparametric estimation of index coefficients. Econometrica 57, 1403-30.

Prakasa Rao, B. L.S. (1983). Nonparametric Functional Estimation. New York: Academic Press.

Racine, J. (1997). Consistent significance testing for nonparametric regression. Journal of Business and Economic Statistics 15, 369-78.

Rahman, M., and A. Ullah (1999). Improved combined parametric and nonparametric regression: Estimation and hypothesis testing. Manuscript, University of California, Riverside.

Rice, J. (1984). Bandwidth choice for nonparametric regression. Annals of Statistics 12, 1215-30.

Robinson, P. M. (1988). Root-n-consistent semiparametric regression. Econometrica 56, 931-54.

Robinson, P. M. (1986). On the consistency and finite sample properties of nonparametric kernel time series regression, autoregression and density estimation. Annals of the Institute of Statistical Mathematics 38, 539-49.

Robinson, P. M. (1989). Nonparametric estimation of time varying parameters. In P. Hack (ed.) Statistical Analysis and Forecasting of Economic Structional Change. Springer-Verlag.

Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. New York: Chapman and Hall.

Speckman, P. (1988). Kernel smoothing in a partial linear model. Journal of Royal Statistical Society Series B 50, 413-46.

Ullah, A. (1985). Specification analysis of econometric models. Journal of Quantitative Economics 1, 187-209.

Ullah, A., and N. Roy (1998). Nonparametric and semiparametric econometrics of panel data. In A. Ullah and D. E.A. Giles (eds.) Handbook of Applied Economic Statistics. ch. 17, pp. 579-604. Marcel Dekker.

Vien, P. (1994). Choice of regressors in nonparametric estimation. Computational Statistics and Data Analysis 17, 575-94.

Watson, G. S. (1964). Smooth regression analysis. Sankhya Series A 26, 359-72.

Zheng, J. X. (1996). Consistent test of functional form via nonparametric estimation techniques. Journal of Econometrics 75, 263 -90.

## Leave a reply